$\newcommand{\vect}[1]{\mathbf{#1}} \newcommand{\uvect}[1]{\hat{#1}} \newcommand{\abs}[1]{\lvert#1\rvert} \newcommand{\norm}[1]{\lVert#1\rVert} \newcommand{\I}{\mathrm{i}} \newcommand{\ket}[1]{\left|#1\right\rangle} \newcommand{\bra}[1]{\left\langle#1\right|} \newcommand{\braket}[1]{\langle#1\rangle} \newcommand{\op}[1]{\mathbf{#1}} \newcommand{\mat}[1]{\mathbf{#1}} \newcommand{\d}{\mathrm{d}} \newcommand{\pdiff}[3][]{\frac{\partial^{#1} #2}{\partial {#3}^{#1}}} \newcommand{\diff}[3][]{\frac{\d^{#1} #2}{\d {#3}^{#1}}} \newcommand{\ddiff}[3][]{\frac{\delta^{#1} #2}{\delta {#3}^{#1}}} \DeclareMathOperator{\erf}{erf} \DeclareMathOperator{\Tr}{Tr} \DeclareMathOperator{\order}{O} \DeclareMathOperator{\diag}{diag} \DeclareMathOperator{\sgn}{sgn} \DeclareMathOperator{\sech}{sech} $